Comodule

In mathematics, a comodule or corepresentation is a concept dual to a module. The definition of a comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebra.

Formal definition

Let K be a field, and C be a coalgebra over K. A (right) comodule over C is a K-vector space M together with a linear map

$\rho\colon M \to M \otimes C$

such that

1. $(\mathrm{id} \otimes \Delta) \circ \rho = (\rho \otimes \mathrm{id}) \circ \rho$
2. $(\mathrm{id} \otimes \varepsilon) \circ \rho = \mathrm{id}$,

where Δ is the comultiplication for C, and ε is the counit.

Note that in the second rule we have identified $M \otimes K$ with $M\,$.

Examples

• A coalgebra is a comodule over itself.

• If M is a finite-dimensional module over a finite-dimensional K-algebra A, then the set of linear functions from A to K forms a coalgebra, and the set of linear functions from M to K forms a comodule over that coalgebra.

• A graded vector space V can be made into a comodule. Let I be the index set for the graded vector space, and let $C_I$ be the vector space with basis $e_i$ for $i \in I$. We turn $C_I$ into a coalgebra and V into a $C_I$-comodule, as follows:

1. Let the comultiplication on $C_I$ be given by $\Delta(e_i) = e_i \otimes e_i$.
2. Let the counit on $C_I$ be given by $\varepsilon(e_i) = 1\$.
3. Let the map $\rho$ on V be given by $\rho(v) = \sum v_i \otimes e_i$, where $v_i$ is the i-th homogeneous piece of $v$.

Rational comodule

If M is a (right) comodule over the coalgebra C, then M is a (left) module over the dual algebra C^∗, but the converse is not true in general: a module over C^∗ is not necessarily a comodule over C. A rational comodule is a module over C^∗ which becomes a comodule over C in the natural way.